In the theory of partial differential equations, a partial differential operator defined on an open subset

is called **hypoelliptic** if for every distribution defined on an open subset such that is (smooth), must also be .

If this assertion holds with replaced by real analytic, then is said to be *analytically hypoelliptic*.

Every elliptic operator with coefficients is hypoelliptic. In particular, the Laplacian is an example of a hypoelliptic operator (the Laplacian is also analytically hypoelliptic). The heat equation operator

(where ) is hypoelliptic but not elliptic. The wave equation operator

(where ) is not hypoelliptic.

## References

- Shimakura, Norio (1992).
*Partial differential operators of elliptic type: translated by Norio Shimakura*. American Mathematical Society, Providence, R.I. ISBN 0-8218-4556-X. - Egorov, Yu. V.; Schulze, Bert-Wolfgang (1997).
*Pseudo-differential operators, singularities, applications*. Birkhäuser. ISBN 3-7643-5484-4. - Vladimirov, V. S. (2002).
*Methods of the theory of generalized functions*. Taylor & Francis. ISBN 0-415-27356-0. - Folland, G. B. (2009).
*Fourier Analysis and its applications*. AMS. ISBN 0-8218-4790-2.

*This article incorporates material from Hypoelliptic on PlanetMath, which is licensed under the Creative Commons Attribution/Share-Alike License.*